In a sentence, if I want to say "the items of my family and I ", can I say it in this way: my family and I's items or is the correct replacement "mine". Can an explanation to why "I's" is correct or incorrect be provided?

In the standard language, "I's" is never correct. A situation where it seems right is a sign that you should rephrase the sentence. (Prescriptively, "of my family and I" is not correct either; it should be "of my family and me", because the pronoun is part of the object of the preposition "of," and in general "I" is a subjective-case pronoun and "me" is an objective-case pronoun. However, this prescription is often not followed so some sources do consider the "and I" construction in objective-case contexts to be de facto part of standard English.)
– sumelicApr 18 '17 at 12:59

There would have to be some unusual context to require our to be broken out into of my family and me.

The possessive of the first person is not formed (and has never been formed) by affixing the -s ending. The possessive form is my.

P.S. Until someone shows me evidence to the contrary, I will continue to regard phrases like "my wife and I's collaboration dinner" (and similar) as the perverse result of prescriptive approaches to teaching grammar in middle school and high school. Throughout the 20th century, schools hammered this rule into the heads of students: "When speaking of yourself and another person, always say "she and I" or "he and I" or "they and I" never "me and her". And what false idea stuck? and I is always the _proper_ second element of such a compound constituent. The speakers are trying to speak properly not naturally, and they have only a fuzzy, distorted memory of what they learned in school.